Comprehensive Study Notes on Functions, Graphing, and Calculator Applications

Definition and Fundamental Properties of Functions

• Functions are analyzed using two variables: an independent variable $x$ and a dependent variable $y$.
  • The dependent variable $y$ relies directly on the value assigned to the independent variable $x$.
  • An example relation expressed algebraically is $y = f(x)$.
• Machine Diagram Analogy:
  • A function can be conceptualized as an input-output machine.
  • You input one value into the machine, and you receive exactly one value as an output.
• Arrow Diagrams (Mapping Diagrams):
  • Represent the domain as a geometric shape (such as a circle or oval) and the range as a separate geometric shape.
  • Directed arrows indicate mappings from specific domain inputs to range outputs.
  • Fundamental Function Criteria: Every single value in the domain must map to only one corresponding value in the range.
• Permissible and Impermissible Mappings:
  • Many-to-One Mapping (Permissible): Multiple distinct elements in the domain can map to the exact same element in the range. This constitutes a valid function.
  • Example: Domain value $1$ maps to range value $20$, and domain value $2$ also maps to range value $20$.
  • Quadratic Example: For the squaring function $y = x^2$, evaluating $x = 2$ yields $y = 4$, and evaluating $x = -2$ also yields $y = 4$. Distinct inputs yield the same output, which is fully valid.
  • One-to-Many Mapping (Impermissible): A single element in the domain maps to multiple distinct elements in the range. This does not constitute a function.
  • Example: Domain value $2$ mapping simultaneously to range value $20$ and range value $3$ is invalid.
  • Purpose of Restriction: Eliminating one-to-many mappings prevents mathematical ambiguity, ensuring that entering an input $x$ produces a single, unambiguous result.

Function Notation, Evaluation, Domain, and Range

• Consider the function:   $f(x) = x^2 + 4$
• Verbal Description and Order of Operations:
  • To evaluate the function verbally, square the input $x$ first (following exponent rules), and then add $4$ to the result.
• Numerical Evaluations:
  • Evaluating at $x = 3$:     $f(3) = 3^2 + 4 = 9 + 4 = 13$
  • Evaluating at $x = -2$:     $f(-2) = (-2)^2 + 4 = 4 + 4 = 8$
  • Note: Enclose negative substitutions in explicit parentheses $(-2)^2$ to guarantee the entire negative quantity is squared to a positive value.
  • Evaluating at $x = \sqrt{5}$:     $f(\sqrt{5}) = (\sqrt{5})^2 + 4 = 5 + 4 = 9$
• Determining Domain:
  • The domain consists of all real number inputs $x$ that produce real outputs.
  • Standard algebraic restrictions to check include division by zero and taking even roots of negative numbers.
  • For $f(x) = x^2 + 4$, no restrictions exist; thus, the domain is all real numbers $(-\infty, \infty)$ or $\mathbb{R}$.
• Determining Range:
  • Squaring any real number yields a non-negative result ($x^2 \ge 0$).
  • The absolute minimum value for $x^2$ is $0$ (occurring at $x = 0$).
  • Adding $4$ shifts the minimum possible output to $0 + 4 = 4$
  • The range of $f(x) = x^2 + 4$ is $[4, \infty)$.

TI-84 Graphing Calculator Setup and Table Operations

• Function Evaluation via Calculator: Consider $f(x) = 3x^2 + x - 5$.
• Primary Key Procedures:
  • Function Entry: Press Y= (top left button directly below screen).
  • Inputting Variable $x$: Use the X,T,\theta,n key located directly to the right of the ALPHA key.
  • Inputting Exponents: Use the square key x^2 or the caret key ^ located above the division sign.
• Window Adjustments (WINDOW button):
  • Xmin and Xmax: Set the left and right horizontal boundaries (e.g., $-10$ to $10$).
  • Ymin and Ymax: Set the lower and upper vertical boundaries.
  • Xscl (X Scale): Sets the distance represented by each tick mark on the horizontal axis.
• Visual Graph Features:
  • A quadratic function containing $x^2$ produces a characteristic U-shaped curve known as a parabola.
  • The presence of the linear term $+ x$ shifts the vertex of the parabola slightly to the left.
• Table Operations (2nd + GRAPH / TABLE):
  • Generates an automated table of input-output pairs.
  • Direct outputs for $f(x) = 3x^2 + x - 5$:
  • At $x = -2$, $f(-2) = 3(-2)^2 + (-2) - 5 = 12 - 2 - 5 = 5$
  • At $x = 0$, $f(0) = 3(0)^2 + 0 - 5 = -5$
  • At $x = 4$, $f(4) = 3(4)^2 + 4 - 5 = 48 + 4 - 5 = 47$
• Table Setup Adjustments (2nd + WINDOW / TBLSET):
  • TblStart: Specifies the initial starting value of $x$ for the table.
  • \Delta Tbl (Delta Table): Specifies the step size or distance increment between consecutive $x$ values.
  • Fractional Step Example: Setting \Delta Tbl = 0.5 enables table evaluation for fractional inputs like $x = 0.5$ (or $\frac{1}{2}$).
  • Output for $x = 0.5$:     $f(0.5) = 3(0.5)^2 + 0.5 - 5 = 3(0.25) + 0.5 - 5 = 0.75 + 0.5 - 5 = -3.7$

Piecewise-Defined Functions

• Definition: A piecewise function applies different algebraic formulas across distinct intervals of its domain.
• Real-World Cell Phone Plan Example:
  • Base cost is $39\,\text{USD}$ per month, which includes up to $2\,\text{GB}$ of data.
  • Data usage exceeding $2\,\text{GB}$ incurs an overage charge of $15\,\text{USD}$ per additional gigabyte.
  • Let $x$ represent data usage in gigabytes, and $C(x)$ represent total monthly cost in dollars.
  • Piecewise Formulation:     $C(x) = \begin{cases} 39 & \text{if } 0 \le x \le 2 \\ 39 + 15(x - 2) & \text{if } x > 2 \end{cases}$
  • Note: The expression $(x - 2)$ calculates the exact data amount exceeding the $2\,\text{GB}$ allowance.
• Sample Evaluations:
  • Evaluating $C(0.5)$: $0.5\,\text{GB}$ falls within $0 \le x \le 2$, so $C(0.5) = 39\,\text{USD}$.
  • Evaluating $C(2)$: $2\,\text{GB}$ falls within $0 \le x \le 2$, so $C(2) = 39\,\text{USD}$.
  • Evaluating $C(4)$: $4\,\text{GB}$ exceeds $2\,\text{GB}$ ($x > 2$):     $C(4) = 39 + 15(4 - 2) = 39 + 15(2) = 39 + 30 = 69\,\text{USD}$

Net Change and Difference Quotients

• Net Change Formula:
  • The net change of a function $f(x)$ between input values $a$ and $b$ (where $b > a$) is defined as:     $\text{Net Change} = f(b) - f(a)$
  • Measures total vertical displacement over the interval $[a, b]$.
• Net Change Example:
  • Given $f(x) = x^2$, find net change from $a = 1$ to $b = 3$:     $\text{Net Change} = f(3) - f(1) = 3^2 - 1^2 = 9 - 1 = 8$
• Algebraic Evaluations for $f(x) = 2x^2 + 3x - 1$:
  • Evaluating $f(a)$: Substitute $a$ for $x$:     $f(a) = 2a^2 + 3a - 1$
  • Evaluating $f(-a)$: Substitute $(-a)$ for $x$:     $f(-a) = 2(-a)^2 + 3(-a) - 1 = 2a^2 - 3a - 1$
  • Evaluating $f(a + h)$: Substitute $(a + h)$ for $x$:     $f(a + h) = 2(a + h)^2 + 3(a + h) - 1$$f(a + h) = 2(a^2 + 2ah + h^2) + 3a + 3h - 1$$f(a + h) = 2a^2 + 4ah + 2h^2 + 3a + 3h - 1$
• Difference Quotient Computation:
  • Calculate $\frac{f(a + h) - f(a)}{h}$:     $\frac{f(a + h) - f(a)}{h} = \frac{(2a^2 + 4ah + 2h^2 + 3a + 3h - 1) - (2a^2 + 3a - 1)}{h}$
  • Distribute negative sign and cancel common opposite terms:     $\frac{2a^2 + 4ah + 2h^2 + 3a + 3h - 1 - 2a^2 - 3a + 1}{h} = \frac{4ah + 2h^2 + 3h}{h}$
  • Factor out $h$ from numerator and cancel with denominator:     $\frac{h(4a + 2h + 3)}{h} = 4a + 2h + 3$
• Applied Astronaut Weight Function:
  • An astronaut weighs $130\,\text{lbs}$ at Earth's surface. Weight $W(h)$ at height $h\,\text{mi}$ above Earth is given by:     $W(h) = 130 \times \left(\frac{3960}{3960 + h}\right)^2$
  • Calculator fraction entry: MATH menu $\rightarrow$ n/d fraction shortcut template.
  • Calculator setup: Set TblStart = 0 and \Delta Tbl = 100.
  • Values:
  • At surface ($h = 0\,\text{mi}$): $W(0) = 130\,\text{lbs}$.
  • At altitude $h = 100\,\text{mi}$: $W(100) \approx 123.67\,\text{lbs}$.

Four Ways to Represent Functions and Graphing Techniques

• Four Modes of Functional Representation:
  1. Verbally: Word description (e.g., "To convert Celsius to Fahrenheit, multiply the Celsius temperature by nine-fifths, then add 32").
  2. Algebraically: Formula notation (e.g., $F(C) = \frac{9}{5}C + 32$).
  3. Numerically: Table of value pairs.
  4. Visually: Graphical plot.
• Calculator Graphing Example: $f(x) = x^3 - 8x^2$
  • Exponent entry: Input $x$, press caret ^, input $3$, and hit right arrow key to exit exponent level.
  • Standard Window Issue: Default window (ZoomStandard, $-10$ to $10$ on both axes) clips graph extrema off screen.
  • Window Adjustment Procedure:
  • Change Xmin to $-2$ to remove empty horizontal space on the left.
  • Keep Xmax at $10$.
  • Set Ymin down to $-30$ (or lower like $-150$) to view full local minimum curve.
  • Set Yscl to $10$ to prevent dense visual clutter from tick marks.

Graphical Analysis: Vertical Line Test, Domain, Range, and Equations

• Vertical Line Test (VLT):
  • A curve in the xy-plane is the graph of a function if and only if no vertical line intersects the curve more than once.
  • Intersects 0 or 1 time: Valid function.
  • Intersects 2 or more times: Not a function (fails VLT because one $x$ input yields multiple $y$ outputs).
• Reading Function Values from a Temperature Graph $T(t)$, where $t$ is hours past noon:
  • Temperature at $t = 1\,\text{hr}$: $T(1) \approx 25^\circ\text{F}$.
  • Temperature at $t = 3\,\text{hrs}$: $T(3) \approx 30^\circ\text{F}$.
  • Temperature at $t = 5\,\text{hrs}$: $T(5) \approx 10^\circ\text{F}$.
  • Comparison: $T(2)$ (approx $35^\circ\text{F}$) is larger than $T(4)$.
  • Solving $T(t) = 25$: Locate horizontal line $y = 25$; yields intersection times $t = 1$ and $t = 4$
  • Solving Inequality $T(t) \ge 25$: Interval $[1, 4]$.
  • Net Change from $t = 1$ to $t = 3$: $T(3) - T(1) = 30 - 25 = 5^\circ\text{F}$.
• Extracting Domain and Range Visually:
  • Domain: Total horizontal extent shadow projected onto $x$-axis.
  • Range: Total vertical extent shadow projected onto $y$-axis.
  • Semi-circle example $f(x) = \sqrt{4 - x^2}$:
  • Domain: $[-2, 2]$
  • Range: $[0, 2]$
• Graphical Solution of Equations and Inequalities:
  • System: Set $Y_1 = 2x^2 + 3$ and $Y_2 = 5x + 6$
  • Calculation procedure: Access CALC menu (2nd + TRACE), select option 5: intersect.
  • Steps: Select first curve (ENTER), second curve (ENTER), and move cursor near desired intersection for Guess (ENTER).
  • Solutions:
  • First intersection point: $(-0.5, 3.5)$
  • Second intersection point: $(3, 21)$
  • Solving $f(x) < g(x)$: Identify intervals where graph $Y_1$ lies strictly below graph $Y_2$.

Increasing, Decreasing, and Local Extrema

• Formal Definitions:
  • Increasing: Function rises left-to-right ($x_1 < x_2 \implies f(x_1) < f(x_2)$).
  • Decreasing: Function falls left-to-right ($x_1 < x_2 \implies f(x_1) > f(x_2)$).
  • Constant: Function remains flat horizontally.
• Interval Reporting Rule:
  • Always write increasing, decreasing, and constant intervals exclusively in terms of $x$ values (domain values).
  • Sample Graph Analysis:
  • Increasing on $(-\infty, -2]$ and $[1, 3]$ (written $(-\infty, -2] \cup [1, 3]$).
  • Constant on $[-2, 1]$.
  • Decreasing on $[3, \infty)$.
• Local Extrema Definitions:
  • Local Maximum: A point $f(a)$ higher than all neighboring points ($f(a) \ge f(x)$ for $x$ near $a$; top of a hill).
  • Local Minimum: A point $f(a)$ lower than all neighboring points ($f(a) \le f(x)$ for $x$ near $a$; bottom of a valley).
• Polynomial Extrema Example: $f(x) = 12x^2 + 4x^3 - 3x^4$
  • Features 2 local maximums and 1 local minimum.
  • Calculating extrema on TI-84: CALC menu $\rightarrow$ 3: minimum or 4: maximum $\rightarrow$ set Left Bound, Right Bound, and Guess.
  • Calculator output note: A display of $1.9999994$ represents numeric approximation rounding to exactly $2$.
  • Local maximum point located at $(2, 3)$.

Classroom Interactions and Calculator Troubleshooting

• Administrative & Course Notes Query:
  • Question: Are physical copies of class note slides provided?
  • Clarification: Notes can be downloaded and printed directly from the course app/portal. Initial first-week copies were printed directly by the instructor.
• Calculator Troubleshooting:
  • Issue: Syntax Error on TI-84 when entering negative values in window setup or formulas.
  • Solution: Use the negative key (-) located on the bottom row next to ENTER, not the subtraction operator - located on the right side key strip.
  • Issue: Table displays fractional outputs instead of decimals.
  • Solution: Dividing out fractions or toggling mode settings converts fractional displays into standard decimal values.